Fractional Power Law Research Articles

Primordial Black Holes in Scalar Field Inflation Coupled to the Gauss–Bonnet Term with Fractional Power-law Potentials

In this study, we investigate the formation of primordial black holes (PBHs) in a scalar field inflationary model coupled to the Gauss–Bonnet term with fractional power-law potentials. The coupling function enhances the curvature perturbations, then results in the generation of PBHs and detectable secondary gravitational waves (GWs). We identify three separate sets of parameters for the potential functions of the form ϕ 1/3, ϕ 2/5, and ϕ 2/3. By adjusting the model parameters, we decelerate the inflaton during the ultra-slow-roll phase and enhance curvature perturbations. Our calculations predict the formation of PBHs with masses of O(10)M⊙ , which are compatible with LIGO-Virgo observational data. Additionally, we find PBHs with masses around O(10−6)M⊙ and O(10−5)M⊙ , which can explain ultra-short-timescale microlensing events in OGLE data. Furthermore, our proposed mechanism could lead to the formation of PBHs in mass scales around O(10−14)M⊙ and O(10−13)M⊙ , contributing to approximately 99% of the dark matter in the Universe. We also study the production of secondary GWs in our model. In all cases of the model, the density parameter of secondary GWs ΩGW0 exhibits peaks that intersect the sensitivity curves of GW detectors, providing a means to verify our findings using data of these detectors. Our numerical results demonstrate a power-law behavior for the spectra of ΩGW0 with respect to frequency, given by ΩGW0(f)∼(f/fc)n . Additionally, in the infrared regime where f ≪ f c , the power index takes a log-dependent form, specifically n=3−2/ln(fc/f) .

Study of quantum Szilard engine for non-interacting bosons in fractional power-law potentials

Abstract In this article, we have realized the quantum Szilard engine (QZE) for non-interacting bosons. We have adopted the Bose-Einstein statistics for this purpose. We have considered fractional power law potential for this purpose and have used the artifact of the quantization of energy. We have calculated the work and the efficiency for non-interacting bosons in fractional power potential. We have shown the dependence of the number of particles for the work and the efficiency. We also have realized the QZE for a single-particle in a Morse potential revealing how the depth of the potential impacts both work and efficiency. Furthermore, we have examined the influence of temperature and the anharmonicity parameter on the work. Finally, we have conducted a comparative analysis, considering both non-interacting bosons in a fractional power law potential and a single-particle in a Morse potential under harmonic approximation conditions.

Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

A Study on Fractional Power-Law Applications and Approximations

The frequency response of the fractional-order power-law filter can be approximated by different techniques, which eventually affect the expected performance. Fractional-order control systems introduce many benefits for applications like compensators to achieve robust frequency and additional degrees of freedom in the tuning process. This paper is a comparative study of five of these approximation techniques. The comparison focuses on their magnitude error, phase error, and implementation complexity. The techniques under study are the Carlson, continued fraction expansion (CFE), Padé, Charef, and MATLAB curve-fitting tool approximations. Based on this comparison, the recommended approximation techniques are the curve-fitting MATLAB tool and the continued fraction expansion (CFE). As an application, a low-pass power-law filter is realized on a field-programmable analog array (FPAA) using two techniques, namely the curve-fitting tool and the CFE. The experiment aligns with and validates the numerical results.

Empirical scaling of formation fracturing by high-energy impulsive mechanical loads

Recent technological advances to trigger high-energy seismic waves from within the wellbore have spurred interest in their potential application to induce fracturing within deep subsurface formations. Although a considerable body of experimental observations at the bench scale (on the order of 1 cubic foot) show promise, there remains considerable uncertainty in how the process may scale to the field. This work characterizes the relationship between the extent and intensity of fracturing and the duration and strength of the dynamic load. Our approach uses a direct numerical simulation of the elastodynamic equations while accounting for nonlinear fracture mechanics. A hybrid Finite-Discrete Element Method (FDEM) is applied whereby cohesive (elastoplastic) laws hold mesh elements together until complete failure. Beyond failure, elements act as deformable free bodies that can interact via contact constraints, including friction. The consistency of the model is first verified and it is validated with observations from previously reported physical experiments. Subsequently, we apply the model to conduct a numerical exploration of the dimensionless parameter space that defines plausible loading waveforms. We model an infinite domain that contains a spherical inclusion within which various impulsive loads are applied. The dynamic loading waveform is parameterized by its rise time to peak stress, followed by a decay period, and occurring within micro- to milliseconds. The dimensionless parameter space is sampled by varying loading characteristics (rise time, peak pressure, and impulse) to reveal the stimulated damaged bulk volume and crack intensity. Consistent with reported experimental observations, the model reproduces a nearly linear trend between the radius of damage and the maximum stress on the bench scale. Beyond that scale, however, the model predicts that this scaling slows considerably to a fractional power law between the damaged radius and the peak stress. This limitation coincides with a geometric increase in the intensity of damage within the stimulated volume. We present loci curves of the expected damage radii as functions of the loading waveform properties.

Impact of Evanescence Process on Three-Dimensional Sub-Diffusion based Molecular Communication Channel.

In most of the existing works of molecular communication (MC), the standard diffusion environment is taken into account where the mean square displacement (MSD) of an information molecule (IM) scales linearly with time. On the contrary, this work considers the sub-diffusion motion that appears in crowded and complex (porous or fractal) environments (movement of the particles in the living cells) where the particle's MSD scales as a fractional order power law in time. Moreover, we examine an additional evanescence process resulting from which the molecules can degrade before hitting the boundary of the receiver (RX). Thus, in this work, we present a 3D MC system with a point transmitter (TX) and the spherical RX with the sub-diffusive behavior of an IM along with its evanescence. Furthermore, an IM's closed-form expressions for the arrival probability and the first passage time density (FPTD) are emulated in the above context. Additionally, we investigate the performance of MC by using the concentration-based modulation technique in a sub-diffusion channel. Finally, the considered MC channel is exploited in terms of the probability of detection, probability of false alarm, and probability of error for different parameters such as the reaction rate, fractional power, and radius of the RX.

Dynamical properties of a meminductor chaotic system with fractal–fractional power law operator

Purpose:This study aims to explore the intricate and concealed chaotic structures of meminductor systems and their applications in applied sciences by utilizing fractal fractional operators (FFOs). Methods:The dynamical analysis of a three-dimensional meminductor system with FFO in the Caputo sense is presented, and a unique solution for the system is obtained via a novel contraction in an orbitally complete metric space. Numerical results are derived using a newly developed method based on Newton polynomial. The analysis includes variations in fractional order and fractal dimension, with a presentation of properties such as Lyapunov spectra, bifurcations, Poincar’e sections, attractor projection, inversion property, and time series analysis. Results:The numerical simulations uncovered intricate hidden attractors for certain values of fractional and fractal orders. This study provides insight into the effects of FFOs on analyzing hidden chaotic attractors and presents a novel approach to understanding and analyzing the dynamics of meminductor systems. The study’s findings contribute to the understanding of the complex and hidden structures of meminductor systems using FFOs. The novel method developed in this study could be applied to other dynamical systems, leading to further advancements in the analysis of complex systems.

Generalizing Fowler–Nordheim Tunneling Theory for an Arbitrary Power Law Barrier

Herein, the canonical Fowler–Nordheim theory is extended by computing the zero‐temperature transmission probability for the more general case of a barrier described by a fractional power law. An exact analytical formula is derived, written in terms of Gauss hypergeometric functions, that fully capture the transmission probability for this generalized problem, including screened interaction with the image potential. First, the quality of approximation against the so far most advanced formulation of Fowler–Nordheim, where the transmission is given in terms of elliptic integrals, is benchmarked. In the following, as the barrier is given by a power law, in detail, the dependence of the transmission probability on the exponent of the power law is analyzed. The formalism is compared with results of numerical calculations and its possible experimental relevance is discussed. Finally, it is discussed how the presented solution can be linked in some specific cases with an exact quantum‐mechanical solution of the quantum well problem.

Fractional-order integral generalization of the Rapoport–Leas equation

A one-dimensional time-fractional integral model for a two-phase flow through a porous medium with power-law memory in the presence of capillary forces is considered. Based on the time-fractional integral generalization of Darcy’s law and the classical mass balance equations for phases, neglecting gravity forces, a time-fractional integral generalization of the Rapoport–Leas equation is derived. It is shown that in the limiting case of constant capillary pressure, the obtained equation coincides with the classical Buckley–Leverett equation. An example of initial boundary value problem for the derived fractional integral equation is given for the case of a finite reservoir. Unlike the classical Rapoport–Leas equation, its time-fractional integral analogue does not have a traveling wave solution in general case. It is due to the fact that the fractional integral with a finite lower limit is not invariant with respect to translations in time. However, for the model problem of two-phase displacement in an unbounded reservoir, considered as a discontinuity breakup problem, the full memory approximation is valid. In this case the obtained equation has a traveling wave solution. In this approximation, relations for the main characteristics of the saturation jump on the shock are obtained. In particular, it is shown that the speed of the shock can be found from an equation that includes the time-fractional integral characteristic of the Leverett function. The behavior of the shock near its left and right boundaries is also asymptotically studied for different saturation levels at the boundaries. It has been proven that if the initial saturation exceeds the residual saturation of the displacing phase, then on the right boundary the saturation at the jump exponentially tends to the initial one. On the left boundary, the asymptotic behavior of the jump is also exponential, with the exception of one special case in which it becomes a fractional power law. For the equation in question, an approximate solution has been analytically constructed for the case of «weak» full memory, when the order of fractional integration is close to zero and can be considered as a small parameter.

Effect of particle shape on packing fraction and velocity profiles at outlet of a silo

Many studies on how the particle shape affects the discharge flow mainly focus on discharge rates and avalanche statistics. In this study, the effect of the particle shape on the packing fraction and velocities of particles in the silo discharge flow are investigated by using the discrete element method. The time-averaged packing fraction and velocity profiles through the aperture are systematically measured for superelliptical particles with different blockinesses. Increasing the particle blockiness is found to increase resistance to flow and reduce the flow rate. At an identical outlet size, larger particle blockiness leads to lower velocity and packing fraction at the outlet. The packing fraction profiles display evidently the self-similar feature that can be appropriately adjusted by fractional power law. The velocity profiles for particles with different shapes obey a uniform self-similar law that is in accord with previous experimental results, which is compatible with the hypothesis of free fall arch. To further investigate the origin of flow behaviors, the packing fraction and velocity field in the region above the orifice are computed. Based on these observations, the flow rate of superelliptical particles is calculated and in agreement with the simulated data.

The Effects of Sheet Thickness and Excitation Frequency on Hysteresis Loops of Non-Oriented Electrical Steel.

The paper focuses on modeling the rate dependence of hysteresis loops in conductive magnetic materials. The concept, which was advanced about fifty years ago by Chua, is discussed. It is shown that the viscous-type equation considered by Zirka and co-workers belongs to the class of Chua-type models. The dynamic effects are described with a simple fractional power law. The value of the exponent in the above-mentioned power law may be assessed on the basis of measurements of coercive field strength at different excitation frequencies. To verify the usefulness of the approach, the measurements of hysteresis loops were carried out at several excitation frequencies under standardized conditions for two grades of non-oriented electrical steel. The modeled curves are in a good correspondence with the measured ones. The considered model uses fewer parameters than approaches based on three-term loss separation schemes.

Fractal Impedance Element Synthesis Based on Multilayer Resistance-Capacitance Environment with C-R-Nc Layer Structure

The idea of fractural calculus appeared almost at the same time as the development of conventional differential calculus began. However, modern scientists became familiar with it only from pioneering works of B. Mandelbrot on fractal geometry of nature in 1980s. Since then, numerous scientific works were published on various aspects of fractural calculus and its application in science and engineering, and the interest to this area remains high. However, for physical implementation of fractural integration operators two-terminal elements with fractional power and frequency law (the fractural impedance elements - FIE) are required. Commercial FIE that could be produced industrially have not been developed so far. The greatest potential, in this case, have FIE built on the basis of multilayer resistance-capacitance element with distributed parameters (RC-distributed elements). One of the baseline designs of such FIEs with R-C-NR layer structure was manufactured as a prototype model. However, when manufacturing and testing of this element, flaws that may prevent its application were detected. The article suggests making FIE based on RC-distributed elements with dual C-R-NC layer structure where, as it is assumed, it is possible to avoid flaws found in FIE with R-C-NR layer structure. However, to solve this problem it is necessary to develop an algorithm and a program of structure synthesis of new type FIE enabling production elements with parameters not worse than those of the existing FIE with R-C-NR layer structure. For this purpose, an algorithm of frequency characteristic analysis of equivalent circuit FIE based on RC-distributed parameters having C-R-NC layer structure and being the component of synthesis algorithm, was developed. Principal stages of synthesis algorithm based on search optimization genetic algorithm were formulated. Ways of information coding of internal structure and parameter C-R-NC FIE were defined. Method of function calculation of individual fitness within population of object representation and ways of genetic operators (selection, crossing-over and mutation) realization in the process of genetic algorithm performance was described. Thereupon the synthesis program was developed and the results of its operation are given in the article. Synthesis result validity was verified by means of circuit simulation software OrCAD by application of Pspice-based C-R-NC-line model. The obtained FIE equivalent circuits integral C-R-NC structures, taking into account chosen materials and layer parameters, were designed. Comparative analysis results of synthesis program operation of C-R-NC FIE and R-C-NR FIE for synthesis FIE with the same requirements to frequency impedance FIE characteristics are given, showing that synthesizing C-R-NC FIE; the FIE with phase uniformity phase-frequency impedance characteristics from - 45° to about - 80° without galvanic coupling between poles and synthesizing R-C-NR FIE, the FIE with phase uniformity of phase-frequency impedance from -10° to about -45° with galvanic coupling between the pole are more likely to be realized.

Self-similarity of density and velocity profiles in a 2D hopper flow of elliptical particles: Discrete element simulation

Two-dimensional discharge flow of elliptical particles with different aspect ratios from a silo is investigated by using discrete element simulation. The measured flow rate of the particles is well fitted via Beverloo's law. The flow rate shows a monotonic decrease with increasing aspect ratio. The density and velocity profiles of particles at the outlet were systematically measured. At an identical outlet size, the speed of particles through the aperture decreases with increasing aspect ratio. The velocity profiles in all cases follow the same definite self-similar law and are in good agreement with the reported experimental results, thereby showing compatibility with the free fall arch postulation. The density profiles show obvious self-similar feature and can be fitted conveniently by using a fractional power law. Based on these observations and the results of the previous work, the flow rate for elliptical particles is calculated and is in agreement with the simulated results.

Temperature dependence of aging dynamics in highly non-equilibrium model polymer glasses.

A central feature of the non-equilibrium glassy "state" is its tendency to age toward equilibrium, obeying signatures identified by Kovacs over 50 years ago. The origin of these signatures, their fate far from equilibrium and at high temperatures, and the underlying nature of the glassy "state" far from equilibrium remain unsettled. Here, we simulate physical aging of polymeric glasses, driven much farther from equilibrium and at much higher temperatures than possible in experimental melt-quenched glasses. While these glasses exhibit Kovacs' signatures of glassy aging at sufficiently low temperatures, these signatures disappear above the onset TA of non-Arrhenius equilibrium dynamics, suggesting that TA demarcates an upper bound to genuinely glassy states. Aging times in glasses after temperature up-jumps are found to obey an Arrhenius law interpolating between equilibrium dynamics at TA and at the start of the temperature up-jump, providing a zero-parameter rule predicting their aging behavior and identifying another unrecognized centrality of TA to aging behavior. This differs qualitatively from behavior of our glasses produced by temperature down-jumps, which exhibit a fractional power law decoupling relation with equilibrium dynamics. While the Tool-Narayanaswamy-Moynihan model can predict the qualitative single-temperature behavior of these systems, we find that it fails to predict the disappearance of Kovacs signatures above TA and the temperature dependence of aging after large temperature up-jumps. These findings highlight a need for new theoretical insights into the aging behavior of glasses at ultra-high fictive temperatures and far from equilibrium.

Design of Pseudo-Random Discrete Signals with Fractional Power-Law Angular Modulation

Design of Pseudo-Random Discrete Signals with Fractional Power-Law Angular Modulation

Investigation of indentation size effect and R-curve behaviour of Li2O–SiO2 and Li2O–2SiO2 glass ceramics

Indentation size effect (ISE) and R-curve behaviour of Li2O-SiO2 and Li2O-2SiO2 glass ceramics are investigated using micro-indentation and indentation-strength (IS) techniques, respectively. Vickers micro-indentations were applied on both materials at the load of 0.10-19.6N to determine the load influence on the measured hardness. For the IS-measured fracture toughness, the load ranged from 1.96 to 19.6N. The hardness decreased with increasing load by 20% and 18% on Li2O-SiO2 and Li2O-2SiO2 glass ceramics, respectively, indicating the ISE behaviour on both materials. The fracture toughness increased with the load by 27% and 59% on Li2O-SiO2 and Li2O-2SiO2 glass ceramics, respectively, signifying the R-curve behaviour. The ISE behaviour of both materials was analysed using the Meyer's, Hays-Kendall (HK), proportional specimen resistance (PSR), Nix-Gao (NG), modified PSR (MPSR) and elastic plastic deformation (EPD) models while the R-curve behaviour was analysed by the fractional power law. The Meyer's index of both materials was less than 2, strongly confirming the ISE existence. The HK, PSR and NG models were only suitable to determine intrinsic Vickers hardness for Li2O-2SiO2 glass ceramic while the MPSR and EPD models were successful for both materials. The fractional power law gave higher R-curve steepness for Li2O-2SiO2 than Li2O-SiO2 glass ceramics. Also, material and brittleness indices predicted, respectively, higher quasi-plasticity and better machinability for Li2O-2SiO2 than Li2O-SiO2 glass ceramics indicating superior performance in the former to the latter. Finally, this study presents a new significant insight into the micro-mechanisms of fracture tolerance behaviour of these glass ceramics which is critical to their functional performance as structural ceramics.

Slip hydrodynamics of combined electroosmotic and pressure driven flows of power law fluids through narrow confinements

Electroosmotic flows (EOF) in microfluidic devices can be greatly enhanced over superhydrophobic surfaces because the high shear rates within the electrical double layer can drive large slip velocities at the interface. Using the power law fluid model, we derive a novel formulation for the Helmholtz–Smoluchowski (HS) velocity and use it to examine the effect of slip on the hydrodynamics of a coupled pressure driven and EOF. Semi analytical relations for the velocity gradient are obtained for cases of a favourable pressure gradient but exact solutions of the velocity can be found only for certain power law indices. Cases of adverse pressure gradient and fractional power law indices are investigated using numerically using the Galerkin Finite Element Method. The validity of the semi analytical relations verified by comparison with the numerical method. The presence of velocity slip at the wall leads to an enhancement of the HS velocity that is most pronounced in shear thinning fluids. Adverse pressure gradients are observed to generate an inflection in the velocity profile and even a two-way flow for certain flow parameters. The strength of the adverse pressure gradient needed to setup a reverse flow at the channel centre reduces as the slip length is increased. The location of the point of inflection is found to depend on the channel height, pressure gradient, electric field, slip length, Debye length and non-Newtonian behaviour.

Multi-level performance-based design optimisation of steel frames with nonlinear viscous dampers

This paper presents a practical multi-level performance-based optimisation method of nonlinear viscous dampers (NVDs) for seismic retrofit of existing substandard steel frames. A Maxwell model is adopted to simulate the behaviour of the combined damper-supporting brace system, with a fractional power-law force–velocity relationship for the NVDs, while a distributed-plasticity fibre-based section approach is used to model the beam-column members thus incorporating the nonlinearity of the parent steel frame in the design process. The optimum height-wise distribution of the damping coefficients of NVDs satisfying given performance requirements is identified via a uniform damage distribution (UDD) design philosophy. The efficiency of the proposed multi-level performance-based design optimisation is illustrated through nonlinear time-history analysis of 3-, 7- and 12-storey steel frames under both artificial and natural spectrum-compatible earthquakes. Sensitivity analysis is performed to investigate the effects of initial height-wise damping distribution, convergence factor and uncertainty in design ground-motion prediction on the optimisation strategy. The efficiency of the final optimum design solution is also investigated by using drift-based, velocity-based, and energy-based UDD approaches to identify the most efficient performance index parameter for optimisation purposes. It is found that regardless of the selected performance parameter, the optimum damping distribution identified by the proposed methodology leads to frames exhibiting lower maximum inter-storey drift, local damage (maximum plastic rotation) and global damage index compared to an equal-cost uniform damping distribution. However, using drift-based UDD approach generally results in a better seismic performance. It is shown that the proposed UDD optimisation method can be efficiently used to satisfy multiple performance objectives at different intensity levels of the earthquake excitation, in line with performance-based design recommendations of current seismic codes. The proposed method is easy to implement for practical design purposes and represents a simple yet efficient tool for optimum seismic retrofit of steel frames with NVDs.

Earthquake dynamic induced by the magma up flow with fractional power law and fractional-order friction.

In this work, we examine the dynamical behaviour of the “single mass-springs” model for earthquake subjected to the strength due to the up flow of magma for the period of volcanism, considering the fractional viscous damping force, the fractional weakening friction and fractional power law of elastic force. The numerical simulation method used in this paper is that of Grünwald-Letnikov based on the generalization of the classical derivative, and the approximately analytical solution obtained by the harmonic balance method. The results have shown that the fractional-order derivative can affect the dynamical properties of fault rock, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude-frequency equation for the steady-state solution was established. It appears that the resonant amplitude and resonant frequency are strongly dependent on the fractional-order damping r, fractional-order friction q, the fractional deflection 𝛼, the nonlinear stiffness coefficient and the fractional viscous coefficient. We have also shown that, the recurrence time of an event, the duration time of an event and the slip size of an earthquake can be controlled by the fractional-order derivative, the fractional-order deflection and the magnitude of the magma strength. The model allowed us to better interpret the earthquake as a stick-slip motion.

Characterization of tissue mimicking phantoms and soft tissues using ultrasound shear wave elastography and power-law models

Characterization of mechanical properties of tissue has become an important area of study over the last two decades. Shear wave elastography (SWE) is being used in many clinical applications to characterize, and multiple mechanical properties are being reported. To more comprehensively characterize soft tissues, we have focused on frequency-domain methods for evaluating viscoelasticity. We measure the shear wave velocity and attenuation and the variation with frequency. We will present results with the Kelvin-Voigt fractional derivative model and general power laws to characterize data from SWE experiments in tissue-mimicking phantoms and exvivo and in vivo tissue. Tissue mimicking phantoms from CIRS, inc., that have varying amounts of viscoelasticity will be investigated. Measurements from ex vivo and in vivo liver and kidney will also be used for this study. We will examine the range of the power-law exponents associated with these samples and the different models to provide understanding of the range of these parameters and how well tissue mimicking phantoms realize the parameters measured in soft tissues.